TECHNICAL REPORT: CVEL-14-066

Modeling the Loading Impedance on Differential-Mode Signals Due to Radiated Emissions

Li Niu1, Todd Hubing1 and Takehiro Takahashi2

1Clemson University, SC, USA 2Takushoku University, Japan

March 1, 2015

Table of Contents

Abstract ..................................................................................................................................................... 3

1. Introduction ....................................................................................................................................... 3

2. Imbalance Difference Theory ........................................................................................................... 4

2.1 Imbalance Factor ........................................................................................................................ 4 2.2 Conversion Impedance ............................................................................................................... 6

3. Experimental Validation ................................................................................................................... 7

3.1 Example Structure and Measurement Setup .............................................................................. 7 3.2 Calculation Procedure ................................................................................................................ 9 3.3 Comparison between Calculation Results and Measurement Results ..................................... 11

4. Conclusion ...................................................................................................................................... 14

References ............................................................................................................................................... 14

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Abstract Imbalance difference theory describes the conversion mechanism between differential-mode signals

and antenna-mode signals on transmission lines. For unintended radiated emission problems, it

provides an easy and yet powerful technique to calculate the antenna-mode current that is converted

from differential-mode signals. In this paper, we introduce conversion impedance to the existing

imbalance difference theory model to account for the loading effect on the differential-mode circuit, so

that when the coupling between differential mode and antenna mode are strong, the imbalance

difference theory can more accurately estimate the AM current.

1. Introduction Unintended radiated emission is a challenging problem for high speed electronic devices; it has

been known for a long time that it is caused by the unintended antenna mode (AM) currents on the

cables or other electrically large metal parts. The AM were frequently referred to in the literatures as

common mode (CM). We, however, distinguish them in the way that the CM exhibits TEM

propagation while the AM does not and it radiates energy away from the structure.

The intended signals on transmission lines are usually differential mode (DM), the fundamental

mechanisms by which differential-mode signals are converted to antenna-mode currents on cables

attached to printed circuit boards were first studied in [1]–[3], where these mechanisms were described

by current-driven models and voltage-driven models. A more precise and easy-to-apply method called

Imbalance Difference Theory (IDT) was introduced later in [4]. IDT pointed out that the unintended

antenna-mode current was generated due to a change in the electrical balance of the transmission lines

carrying the differential signal currents. The exact antenna-mode current can be accurately calculated

based on the transmission line geometries and the strength of the differential-mode signal at the

interface.

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The IDT has been successfully applied to a number of radiated emission problems since its

introduction [5]–[17]. However it wasn’t rigorously derived until a recently published paper by the

author [21], where we demonstrated that if the imbalance factor is defined as the actual current

division factor, IDT is strictly correct for radiated emission calculation. We also shown in that paper

the conventional method of calculating imbalance factor by analyzing the cross sections of the

transmission lines [4] [8] was a very close approximation to the actual current division factor.

For radiated emission problems, the DM and AM signals are usually weakly coupled: only a small

portion of the DM energy is converted to AM energy, and the energy converted back to DM is even

smaller and can be neglected. For the strong coupling case however, ignoring the energy converted

back to DM can affect the accuracy of the calculation. In another recently published paper [22], the

IDT was applied to a multi-conductor transmission line structure, where the CM signals exhibit TEM

propagation and the DM and CM signals are strongly coupled. We introduced the concept of

conversion impedance to the IDT model to account for the loading effect to the original DM circuit

due to DM-CM conversion.

In this paper we first explore the conversion impedance of IDT model for radiation emission

applications. Then we provide an example calculation to show that when the coupling between DM

and AM is strong, i.e. when the structure hit resonant frequency, the conversion impedance has big

impact over the accuracy of IDT model.

2. Imbalance Difference Theory 2.1 Imbalance Factor

Consider a two-conductor transmission line with a cross-section that suddenly changes as shown in

Fig. 1. At the interface where the cross-section changes, the voltage between the two conductors is VDM.

As described in [21], the change in the electrical balance of the conductors results in an antenna-mode

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voltage that drives the conductors on one side of the interface relative to the conductors on the other

side of the interface as indicated in Fig. 2. The amplitude of the driving voltage is given by,

( )AM DMV h V= ∆ ⋅ . (1)

where Δh is the change in the imbalance factor occurring at the interface and VDM is the differential-

mode voltage at the interface. The imbalance factor of each section of the transmission lines is defined

as the ratio of the AM currents on each conductor in Fig. 2.

Fig. 1. Two-conductor transmission lines with changed cross-section.

Fig. 2. Antenna mode of TLs with divided AM current.

The imbalance factor for left and right section of the TL, hL and hR are:

1 1 1/ ( )L AM L AM L AM Lh I I I− − −= + . (2)

1 1 1/ ( )R AM R AM R AM Rh I I I− − −= + . (3)

The IAM-1L, IAM-2L, IAM-1R, and IAM-1R denote the AM current on each conductor of the TLs in Fig. 2.

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For realistic radiated emission applications, the current division factor or imbalance factor is hard to

obtain precisely, but [21] demonstrated that expressing h as a ratio of the per-unit-length inductances

or capacitances is a very good approximation. The equations are:

11

11 22

ChC C

=+

. (4)

22 12

11 22 122L Lh

L L L−

=+ −

. (5)

Detailed definitions of C11, C22, L11, L22 and L12 can be found in [8].

2.2 Conversion Impedance The imbalance difference theory describes a method to calculate the conversion between DM and

CM/AM signals [4] [8]. It points out that the conversion from one mode to the other is due to the

change of electrical balance along the transmission lines and the strength of the conversion is

proportional to the change of the imbalance factor.

Based on IDT, for the circuit in Fig. 1, the generated AM current will be equal to that in Fig. 2

when the AM voltage source is:

( )AM DM R LV V h h= ⋅ − . (6)

If we denote the input impedance of the antenna that the AM voltage source sees in Fig. 2 as ZAM, then

the AM current is:

/ ( ) /AM AM AM DM R L AMI V Z V h h Z= = ⋅ − . (7)

According to the IDT, at the interface where imbalance factor changes, there will also be conversion

from AM current to DM current as:

2( ) ( ) /DM AM R L DM R L AMI I h h V h h Z∆ = − ⋅ − = − ⋅ − . (8)

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The extra DM current, ΔIDM, virtually flows from one conductor to the other at the interface, as

shown in Fig. 3. Its effect over the DM circuit can be represented by an impedance, which we call

conversion impedance, ZDA, as shown in Fig. 4.

Fig. 3. DM circuit with the extra virtual DM current.

Fig. 4. DM circuit with the conversion impedance.

The expression for the conversion impedance is:

2| | ( )DM AM

DADM R L

V ZZI h h

= =∆ −

. (9)

3. Experimental Validation 3.1 Example Structure and Measurement Setup

As shown in Fig. 5, we connected a twisted wire pair (TWP) to a coaxial cable and kept the

structure standing vertically on a metal ground plane. The structure was fed by a DM voltage through

underground coaxial cable. The change of electrical balance at the interface between the coaxial cable

and the TWP produces AM current. The AM current was measured at the bottom of the antenna close

to the ground surface. The feeding DM voltage was measured by an oscilloscope through a T-

connector.

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Fig. 5. Validation structure and measurement set up. Fig. 6 is a photo of the test setup. The tested TWP and coaxial cable were placed in a semi-anechoic

chamber and the coaxial cables used for feeding and measurement were placed close to the ground

plane so that they had very little effect on the antenna. Measurement set-up parameters are listed in

Table. 1.

Fig. 6. Validation structure and measurement set up.

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Table. 1. Parameters of measurement setup.

TWP wire AWG 18 (conductor diameter: 1mm, insulator thickness 0.75mm)

Coaxial cable RG-58AU

Standing coaxial cable length 0.5 meter

TWP length 0.5 meter

Length of coaxial cable from the feeding point to the T-connection 1.41 meter

Signal generator BK precision 4087

Oscilloscope Tektronix MSO 4104

Current probe Fisher F-33-1

Measured wave velocity of propagation in the TWP

82.0 10 /m s⋅

TWP termination Open circuit

3.2 Calculation Procedure The structure in Fig. 5 can be modeled as the circuit shown in Fig. 7. The goal of the first part of the

calculation is to determine the DM voltage at the interface where the TWP and coaxial cable connect

so that we can apply IDT to calculate the equivalent AM voltage source that drives the TWP-coaxial-

cable antenna.

Fig. 7. Equivalent circuit for measurement set up.

The characteristic impedance of the twisted wire pair, ZDM-TWP can be calculated by:

120 2ln( )DM TWPr

sZdε−

⋅= ⋅ . (10)

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where εr is the relative permittivity of the insulation material; s is the distance between the centers of

two wires, d is the diameter of the conductor in the wires.

ZDM-Coax, ZOSC, and ZSG are the characteristic impedance of the coaxial cable, the input impedance of

the oscilloscope and the output impedance of the signal generator, respectively. They are all 50ohms.

The twisted wire pair is a typical balanced transmission line and the imbalance factor, hTWP, is 0.5. The

coaxial cable is a typical perfectly unbalanced transmission line and the imbalance factor, hCoax, is 1.

The change of imbalance factor at the interface where TWP connects to the coaxial cable is,

0.5Coax TWPh h h∆ = − = . (11)

The ZDA in Fig. 7 is the DM-to-AM conversion impedance. It can be calculated using (9). The input

impedance of the antenna was calculated using the antenna modeling software, 4NEC2 [18], where

solid wires were used to represent the TWP and the coaxial cable. The equivalent radius used for the

coaxial cable was the same as the cable-shield’s radius; the one used for the TWP was calculated as

[19]:

/ 2TWPR s d= ⋅ . (12)

The input impedance looking into the TWP from the interface is,

1tan( )in TWP DM TWP

TWP TWP

Z Zj lβ− −= ⋅⋅ ⋅

, (13)

where lTWP is the length of the TWP and βTWP is the phase constant of the TWP.

The load impedance that the coaxial cable sees at the interface is,

L mid DA in TWPZ Z Z− −= . (14)

The input impedance looking into the coaxial cable is,

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tan( )tan( )

L mid DM Coax Coax Coaxin Coax DM Coax

DM Coax L mid Coax Coax

Z j Z lZ ZZ j Z l

ββ

− −− −

− −

+ ⋅ ⋅ ⋅= ⋅

+ ⋅ ⋅ ⋅ . (15)

where βcoax is the phase constant of the coaxial cable and the lCoax is the length of the coaxial cable

from the interface where it connects to the TWP to the T-connection.

The DM voltage at the T-connection that feeds the coaxial cable can be calculated as,

in Coax OSCcoax feed SG

in Coax OSC SG

Z ZV VZ Z Z

−−

−

= ⋅+

. (16)

The reflection coefficient at the interface where coaxial cable connects to TWP is,

L mid DM Coaxmid

L mid DM Coax

Z ZZ Z

− −

− −

−Γ =

+ . (17)

So the positive propagation voltage at the interface is,

coax coax coax coax

coax feedmid j l j l

VV

e eβ β−+

⋅ ⋅ − ⋅ ⋅=+ Γ ⋅

. (18)

and the DM voltage at the interface is,

DM mid midV V V+ += + ⋅Γ . (19)

Applying IDT yields the equivalent AM voltage source that drives the antenna:

AM DMV V h= ⋅∆ . (19)

With the equivalent AM voltage source, we build the structure similar as that in 4nec2 in a MOM

simulation software, FEKO [20], to calculate the AM current at the bottom of the antenna.

3.3 Comparison between Calculation Results and Measurement Results We calculated the DM voltage that feeds the coaxial cable at the T-connection in Fig. 5 and the AM

current at the bottom of the antenna. The IDT was applied both with and without conversion

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impedance ZDA. The comparisons of the calculated results and the measurement results are shown in

Fig. 8 and Fig. 9.

Fig. 8. Comparison of DM voltage at the T-connection.

Fig. 9. Comparison of AM current at the bottom of the antenna.

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We can see from Fig. 8 and Fig. 9 that the calculated results with conversion impedance, labelled as

“Calculated” in the plots, are very close to the measurement result over the frequency from 30MHz to

100MHz. The results without conversion impedance, noted as “old-Calculated” in both plots, are very

close to that with conversion impedance over most of the frequency range except at resonant frequency,

around 70MHz, where the old model over-estimated the AM current.

Here is the explanation: According to (9), the conversion impedance is proportional to the input

impedance of the antenna, ZAM , in our case, it is 4 times of ZAM. At non-resonant frequencies, ZAM is

about couple hundreds ohms, which can be seen on an input-impedance-over-frequency plot on Fig. 10,

so the ZDA is much bigger than the DM impedances. At resonant frequency, however, the input

impedance is about 70 ohms and the ZDA is less than 300 ohms, which is comparable to the DM

impedances, as a result neglecting ZDA causes less accurate calculation results.

Fig. 10. Input impedance of the antenna seen by the AM voltage source.

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4. Conclusion In this paper we introduce the conversion impedance to the imbalance difference theory for

modelling radiated emissions. For most practical radiating structures, at non-resonant frequencies, the

conversion impedance is much larger than the DM impedances in the circuit and it has little impact

over the accuracy of IDT models. However, we demonstrate with an example structure that the

conversion impedance can have big influence over the accuracy of the IDT model if the radiating

structure hits resonant frequency and the conversion impedance becomes comparable to the DM

impedances.

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Radiated Emission Problems,” CVEL-14-056, Dec. 20, 2014. [22] L. Niu and T. Hubing, “Modeling the Conversion between Differential Mode and Common Mode

Propagation in Transmission Lines,” CVEL-14-055, Mar. 1, 2015.